On the Thomas-Fermi ground state in a harmonic potential

TL;DR

This paper rigorously justifies the Thomas-Fermi approximation for nonlinear ground states of the Gross-Pitaevskii equation in harmonic potentials across different dimensions, using Painlevé-II equations, and characterizes eigenvalue distributions in 1D.

Contribution

It provides a new uniform spatial scale justification of the Thomas-Fermi approximation and characterizes eigenvalues in the 1D case, advancing understanding of nonlinear quantum states.

Findings

Thomas-Fermi approximation justified on a uniform spatial scale

Eigenvalue distribution characterized in 1D

Extension of variational methods to nonlinear ground states

Abstract

We study nonlinear ground states of the Gross-Pitaevskii equation in the space of one, two and three dimensions with a radially symmetric harmonic potential. The Thomas-Fermi approximation of ground states on various spatial scales was recently justified using variational methods. We justify here the Thomas-Fermi approximation on an uniform spatial scale using the Painlev\'{e}-II equation. In the space of one dimension, these results allow us to characterize the distribution of eigenvalues in the point spectrum of the Schr\"{o}dinger operator associated with the nonlinear ground state.